Nemytskii operator

In mathematics, Nemytskii operators are a class of nonlinear operators on L^p spaces with good continuity and boundedness properties. They take their name from the mathematician Viktor Vladimirovich Nemytskii.

General definition of Superposition operator

Let $𝕏,𝕐,\mathbb{Z}\ne \varnothing$ be non-empty sets. Let ${𝕐}^{𝕏},{\mathbb{Z}}^{𝕏}$ denote the sets of mappings from $𝕏$ to $𝕐$ and $\mathbb{Z}$ respectively. Let $h:𝕏\times 𝕐\to \mathbb{Z}$.

Then the Nemytskii superposition operator $H:{𝕐}^{𝕏}\to {\mathbb{Z}}^{𝕏}$ induced by $h$ is the map taking any map $\phi \in {𝕐}^{𝕏}$ to the map $H\phi \in {\mathbb{Z}}^{𝕏}$ defined by $$(H\phi )(x)=h(x,\phi (x))\in \mathbb{Z},\text{for all}x\in 𝕏.$$ The function $h$ is called the generator of the Nemytskii operator $H$.

Definition of Nemytskii operator

Let Ω be a domain (an open and connected set) in n-dimensional Euclidean space. A function f : Ω × R^m → R is said to satisfy the Carathéodory conditions if

• f(x, u) is a continuous function of u for almost all x ∈ Ω;
• f(x, u) is a measurable function of x for all u ∈ R^m.

Given a function f satisfying the Carathéodory conditions and a function u : Ω → R^m, define a new function F(u) : Ω → R by

$F(u)(x)=f(x,u(x)).$

The function F is called a Nemytskii operator.

Theorem on Lipschitzian Operators

Suppose that $h:[a,b]\times ℝ\to ℝ$, $X=\text{Lip}[a,b]$ and

$$H:\text{Lip}[a,b]\to \text{Lip}[a,b]$$

where the operator $H$ is defined as $\left(Hf\right)\left(x\right)$ $=h(x,f(x))$ for any function $f:[a,b]\to ℝ$ and any $x\in [a,b]$. Under these conditions the operator $H$ is Lipschitz continuous if and only if there exist functions $G,H\in \text{Lip}[a,b]$ such that

$$h(x,y)=G(x)y+H(x),x\in [a,b],y\in ℝ.$$

Boundedness theorem

Let Ω be a domain, let 1 < p < +∞ and let g ∈ L^q(Ω; R), with

$\frac{1}{p}+\frac{1}{q}=1.$

Suppose that f satisfies the Carathéodory conditions and that, for some constant C and all x and u,

$|f(x,u)|\le C|u{|}^{p-1}+g(x).$

Then the Nemytskii operator F as defined above is a bounded and continuous map from L^p(Ω; R^m) into L^q(Ω; R).

References

• Template:Cite book (Section 10.3.4)

• Template:Cite journal